launch your bitcoin client as usual and wait for it to load the blockchain and start up
click on 'help' in the menu bar (top right)
click on 'debug window'
select the 'console' tab
type: walletpassphrase "your walletpassphrase here" 600
type: dumpprivkey [your public key here]
this will return the private key, you can copy it now; ensure you clear your clipboard/history afterwards
type: walletlock

I do not understand what "your walletpassphrase here" means. I do not have a wallet pass phrase to the best of my knowledge.

Also, I am a bit of an amateur at this and I still don't understand the "your public key here"..... is that the address that corresponds to your wallet?

Trying to export the private key to another wallet because the bitcoin wallet has taken 2 days to synch and is still not done.

You shouldn't be touching keys. This is something that only someone debugging the wallet software should do, and you risk compromising your security and/or losing your bitcoins if you try to mess with them!
– Luke-JrDec 14 '14 at 4:18

This is not good enough, as I found today. I believe this does not list internal change addresses that bitcoind creates when sending outgoing transactions. Starting from bitcoind 0.7, you can use listaddressgroupings, which lists all addresses.
– ripper234Oct 19 '13 at 14:15

When I was trying to figure this out I found a list of all Bitcoin console commands. It's below for your reference. All of the console commands can be executed by opening up your Bitcoin wallet software, click on help, then Console.